LING/C SC/PSYC 438/538 Computational Linguistics Sandiway Fong Lecture 11: 10/2

Last Time FSA –gave a formal definition (Q,s,f,Σ,  ) –background reading Chapter 2: Regular Expressions and Finite State Automata –many practical applications can be encoded and run efficiently on a computer implement regular expressions compress large dictionaries build morphological analyzers (suffixation) –see chapter 3 of textbook speech recognizers –(Hidden) Markov models = FSA + probabilities

FSA: Recap formally –(Q,s,f,Σ,  ) 1.set of states (Q): {s,w,x,y,z} 4 statesmust be a finite set 2.start state (s): s 3.end state(s) (f): z 4.alphabet ( Σ ): {a, b, !} 5.transition function  : signature: character × state → state  (b,s)=w  (a,w)=x  (a,x)=y  (a,y)=y  (!,y)=z sw b x a a > y a z ! xy a λ λ (or ε ) denotes the empty transition

regexps and FSA: recap anything we can encode with a regular expression, we can build a FSA for it basic operators, e.g. –+–+ one or more of the preceding character e.g. a+ backreferences: –by state splitting –e.g. ([aeiou])\1 xy o a e i u yz o a e i u x yuyu o a e i u z yaya yeye yiyi yoyo o a e u xy a a

Today’s Lecture how to implement FSA –in Prolog –two ways: one based directly on the formal definition...

Finite State Automata (FSA) formally –(Q,s,f,Σ,  ) 1.Q: {s,x,y} 2.s: s 3.f: y 4.Σ : {a, b} 5.transition function  : signature: character × state → state  (a,s)=x  (a,x)=x  (b,x)=y  (b,y)=y sx y a a b b >

Finite State Automata (FSA) directly implement the formal definition –define a predicate fsa/2 –takes two arguments –S = a start state –L = string (as a list) we’re interested in testing Prolog code (for any FSA) 1.fsa(S,L) :- L = [C|M], transition(S,C,T), fsa(T,M). 2.fsa(E,[]):- end_state(E). :- to be read as “true if”

Finite State Automata (FSA) Prolog code (for any FSA) –fsa(S,L) :- L = [C|M], transition(S,C,T), fsa(T,M). –fsa(E,[]):- end_state(E). Facts (FSA-particular) –end_state(y). –transition(s,a,x). –transition(x,a,x). –transition(x,b,y). –transition(y,b,y). sx y a a b b transition function  :  (a,s)=x  (a,x)=x  (b,x)=y  (b,y)=y >

Finite State Automata (FSA) –computation tree ?- fsa(s,[a,a,b]). ?- transition(s,a,T). T=x ?- fsa(x,[a,b]). ?- transition(x,a,T’). T’=x ?- fsa(x,[b]). ?- transition(x,b,T”). T”=y ?- fsa(y,[]). ?- end_state(y). Yes fsa(S,L) :- L = [C|M], L = [C|M],transition(S,C,T),fsa(T,M). fsa(y,[]) :-. fsa(y,[]) :- end_state(E).

Finite State Automata (FSA) deterministic FSA (DFSA) –no ambiguity about where to go at any given state non-deterministic FSA (NDFSA) –no restriction on ambiguity (surprisingly, no increase in formal power) textbook –D-RECOGNIZE (FIGURE 2.13) –ND-RECOGNIZE (FIGURE 2.21) fsa(S,L) :- L = [C|M], L = [C|M],transition(S,C,T),fsa(T,M). fsa(y,[]) :-. fsa(y,[]) :- end_state(E).

Finite State Automata (FSA) Prolog Advantages –no change in code for NDFSA –Prolog computation rule takes care of choice point management example –one change –“a” instead of “b” from x to y –non-deterministic –what regular language does this machine accept? sx y a a b a

Finite State Automata (FSA) another possible Prolog encoding strategy –define one predicate for each state taking one argument (the input string) consume input character call next state with remaining input string –query ?- s(L). call start state s

Finite State Automata (FSA) –state s: (start state) s([a|L]) :- x(L). match input string beginning with a and call state x with remainder of input –state x: x([a|L]) :- x(L). x([b|L]) :- y(L). –state y: (end state) y([]). y([b|L]) :- y(L). sx y a a b b

Finite State Automata (FSA) example: 1.?- s([a,a,b]). 2.?- x([a,b]). 3.?- x([b]). 4.?- y([]). Yes s([a|L]) :- x(L). x([a|L]) :- x(L). x([b|L]) :- y(L). y([]). y([b|L]) :- y(L).

Finite State Automata (FSA) Note: –non-deterministic properties of Prolog’s computation rule still applies here

Finite State Automata (FSA) example 1.?- s([a,b,a]). 2.?- x([b,a]). 3.?- y([a]). No s([a|L]) :- x(L). x([a|L]) :- x(L). x([b|L]) :- y(L). y([]). y([b|L]) :- y(L).

Class Project Implement the following FSA in Prolog –using the one predicate per state model sxy a a b λ > make sure your program can generate